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This article aims to establish a semi-analytical approach based on thehomotopy perturbation method (HPM) to find the closed form or approximatedsolutions for the population balance equations such as Smoluchowski'scoagulation, fragmentation, coupled coagulation-fragmentation and bivariatecoagulation equations. An accelerated form of the HPM is combined with theElzaki transformation to improve the accuracy and efficiency of the method. Oneof the significant advantages of the technique lies over the classic numericalmethods as it allows solving the linear and non-linear differential equationswithout discretization. Further, it has benefits over the existingsemi-analytical techniques such as Adomian decomposition method (ADM),optimized decomposition method (ODM), and homotopy analysis method (HAM) in thesense that computation of Adomian polynomials and convergence parameters arenot required. The novelty of the scheme is shown by comparing the numericalfindings with the existing results obtained via ADM, HPM, HAM and ODM fornon-linear coagulation equation. This motivates us to extend the scheme forsolving the other models mentioned above. The supremacy of the proposed schemeis demonstrated by taking several numerical examples for each problem. Theerror between exact and series solutions provided in graphs and tables show theaccuracy and applicability of the method. In addition to this, convergence ofthe series solution is also the key attraction of the work.

Elzaki Transform Based Accelerated Homotopy Perturbation Method for  Multi-dimensional Smoluchowski's Coagulation and Coupled  Coagulation-fragmentation Equations